A priori convergence of the Greedy algorithm for the parametrized reduced basis method

Annalisa Buffa (1) , Yvon Maday (2) , Anthony T. Patera (3) , Christophe Prud'Homme (4, 5) , Gabriel Turinici (6)

1 UNIPV - Dipartimento di Matematica “Felice Casorati” Department of Mathematics [Univ Pavia]
2 LJLL - Laboratoire Jacques-Louis Lions
3 MIT-MECHE - Department of Mechanical Engineering [Massachusetts Institute of Technology]
4 IRMA - Institut de Recherche Mathématique Avancée
5 EDP - Equations aux Dérivées Partielles
6 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

ABSTRACT The convergence and efficiency of the reduced basis method used for the approximation of the solutions to a class of problems written as a parametrized PDE depends heavily on the choice of the elements that constitute the "reduced basis". The purpose of this paper is to analyze the a priori convergence for one of the approaches used for the selection of these elements, the greedy algorithm. Under natural hypothesis on the set of all solutions to the problem obtained when the parameter varies, we prove that three greedy algorithms converge; the last algorithm, based on the use of an a posteriori estimator, is the approach actually employed in the calculations.

Mots clés: Greedy algorithm; reduced basis approximations; a priori analysis; best fit analysis

Domaines: Analyse numérique [math.NA]

Annalisa Buffa, Yvon Maday, Anthony T. Patera, Christophe Prud'Homme, Gabriel Turinici. A priori convergence of the Greedy algorithm for the parametrized reduced basis method. ESAIM: Mathematical Modelling and Numerical Analysis, 2012, 46 (3), pp.595-603. ⟨10.1051/m2an/2011056⟩. ⟨hal-00659314⟩